Satake diagram

In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by Satake (1960, p.109) whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a Dynkin diagram classify real forms of the complex Lie algebra corresponding to the Dynkin diagram.

More generally, the Tits index or Satake–Tits diagram of a reductive algebraic group over a field is a generalization of the Satake diagram to arbitrary fields, introduced by Tits (1966), that reduces the classification of reductive algebraic groups to that of anisotropic reductive algebraic groups.

Satake diagrams are not the same as Vogan diagrams of a Lie group, although they look similar.

Definition

A Satake diagram is obtained from a Dynkin diagram by blackening some vertices, and connecting other vertices in pairs by arrows, according to certain rules.

Suppose that G is an algebraic group defined over a field k, such as the reals. We let S be a maximal split torus in G, and take T to be a maximal torus containing S defined over the separable algebraic closure K of k. Then G(K) has a Dynkin diagram with respect to some choice of positive roots of T. This Dynkin diagram has a natural action of the Galois group of K/k. Also some of the simple roots vanish on S. The Satake–Tits diagram is given by the Dynkin diagram D, together with the action of the Galois group, with the simple roots vanishing on S colored black. In the case when k is the field of real numbers, the absolute Galois group has order 2, and its action on D is represented by drawing conjugate points of the Dynkin diagram near each other, and the Satake–Tits diagram is called a Satake diagram.

Examples

• Compact Lie algebras correspond to the Satake diagram with all vertices blackened.
• Split Lie algebras correspond to the Satake diagram with only white (i.e., non blackened) and unpaired vertices.
• A table can be found at (Onishchik & Vinberg 1994, Table 4, pp. 229–230).

Differences between Satake and Vogan diagrams

Both Satake and Vogan diagrams are used to classify semisimple Lie groups or algebras (or algebraic groups) over the reals and both consist of Dynkin diagrams enriched by blackening a subset of the nodes and connecting some pairs of vertices by arrows. Satake diagrams, however, can be generalized to any field (see above) and fall under the general paradigm of Galois cohomology, whereas Vogan diagrams are defined specifically over the reals. Generally speaking, the structure of a real semisimple Lie algebra is encoded in a more transparent way in its Satake diagram, but Vogan diagrams are simpler to classify.

The essential difference is that the Satake diagram of a real semisimple Lie algebra ${\mathfrak {g}}$ with Cartan involution θ and associated Cartan pair ${\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}$ (the +1 and −1 eigenspaces of θ) is defined by starting from a maximally noncompact θ-stable Cartan subalgebra ${\mathfrak {h}}$, that is, one for which $\theta ({\mathfrak {h}})={\mathfrak {h}}$ and ${\mathfrak {h}}\cap {\mathfrak {k}}$ is as small as possible (in the presentation above, ${\mathfrak {h}}$ appears as the Lie algebra of the maximal split torus S), whereas Vogan diagrams are defined starting from a maximally compact θ-stable Cartan subalgebra, that is, one for which $\theta ({\mathfrak {h}})={\mathfrak {h}}$ and ${\mathfrak {h}}\cap {\mathfrak {k}}$ is as large as possible.

The unadorned Dynkin diagram (i.e., that with only white nodes and no arrows), when interpreted as a Satake diagram, represents the split real form of the Lie algebra, whereas it represents the compact form when interpreted as a Vogan diagram.

See also

• Relative root system
• List of irreducible Tits indices

References

• Bump, Daniel (2004), Lie groups, Graduate Texts in Mathematics, vol. 225, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-4094-3, ISBN 978-0-387-21154-1, MR 2062813
• Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, Providence, R.I.: American Mathematical Society, doi:10.1090/gsm/034, ISBN 978-0-8218-2848-9, MR 1834454
• Onishchik, A. L.; Vinberg, Ėrnest Borisovich (1994), Lie groups and Lie algebras III: structure of Lie groups and Lie algebras, Springer, ISBN 978-3-540-54683-2
• Satake, Ichirô (1960), "On representations and compactifications of symmetric Riemannian spaces", Annals of Mathematics, Second Series, 71 (1): 77–110, doi:10.2307/1969880, ISSN 0003-486X, JSTOR 1969880, MR 0118775
• Satake, Ichiro (1971), Classification theory of semi-simple algebraic groups, Lecture Notes in Pure and Applied Mathematics, vol. 3, New York: Marcel Dekker Inc., ISBN 978-0-8247-1607-3, MR 0316588
• Spindel, Philippe; Persson, Daniel; Henneaux, Marc (2008), "Spacelike Singularities and Hidden Symmetries of Gravity", Living Reviews in Relativity, 11 (1): 1, arXiv:0710.1818, Bibcode:2008LRR....11....1H, doi:10.12942/lrr-2008-1, PMC 5255974, PMID 28179821
• Tits, Jacques (1966), "Classification of algebraic semisimple groups", Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Providence, R.I.: American Mathematical Society, pp. 33–62, MR 0224710
• Tits, Jacques (1971), "Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque", Journal für die reine und angewandte Mathematik, 1971 (247): 196–220, doi:10.1515/crll.1971.247.196, ISSN 0075-4102, MR 0277536, S2CID 116999784